We consider the geometrically and physically nonlinear theory of elastic rods (Cosserat – Timoshenko rods) and the elastic Cosserat medium. These theories consider independent translational and rotational degrees of freedom. Systems of differential equations for these theories are well known. In this paper we obtain the variational formulation of static problems for these theories. The variational problem was defined as a problem of searching for the stationary points of functionals. The variational functionals are important for the correct formulation of finite element method algorithms for nonlinear problems, as well as for the formulation and solution of stability problems.